Loading...
Una topología fuertemente conexa sobre los naturales, su generalización sobre dominios integrales y aplicaciones
Macías, Jhixon
Macías, Jhixon
Citations
Altmetric:
Abstract
Motivado por los estudios sobre los espacios topológicos de Furstenberg y Golomb, el autor definió la topología $\tau_M$ sobre el conjunto de los enteros positivos $\mathbb{N}$, la cual es generada por la base $\beta_M:=\{\sigma_n: n\in\mathbb{N}\}$ donde $\sigma_n:=\{m\in\mathbb{N}: \gcd(n,m)=1\}$. La topología $\tau_M$ puede ser generalizada sobre anillos conmutativos $R$ con $1_R\neq 0_R$ como sigue: para cada $r\in R\setminus\{0\}$ se define $\sigma_r:=\{s\in R\setminus\{0\}:\langle r\rangle+\langle s \rangle=R\}$. Entonces, la generalización de la topología $\tau_M$ sobre $R\setminus\{0\}$ es la topología $\tau_{R^0}$ generada por la base $\beta_R:=\{\sigma_r: r\in R\setminus\{0\}\}$. En este trabajo, se estudió la topología $\tau_M$ y su generalización sobre dominios integrales. Tanto en $\mathbb{N}$, como sobre cualquier dominio integral $R$, los espacios topológicos $(\mathbb{N},\tau_M)$ y $(R\setminus\{0\},\tau_{R^0})$ son hiperconexos, ultraconexos, y más gruesos que los espacios de Golomb. En particular, cuando $R$ es un cuerpo, el espacio topológico $(R\setminus\{0\},\tau_{R^0})$ es el trivial. Sobre $\mathbb{N}$ se demuestra que el conjunto de funciones polinomiales continuas sobre $(\mathbb{N},\tau_M)$ son los monomios sobre $\mathbb{N}$. También se demuestra, que el conjunto de homeomorfismos sobre $(\mathbb{N},\tau_M)$ es el conjunto de biyecciones $h$, sobre $\mathbb{N}$, tales que para todo entero positivo $n$, se tiene que $h(\sigma_n)=\sigma_{h(n)}$. Se presenta una prueba topológica de la infinitud de los números primos y de la infinitud de los números primos en la progresión aritmética $4(\mathbb{N}\cup\{0\})+3$. Sobre dominios integrales infinitos, que no son cuerpos y cuya cardinalidad es más grande que la del conjunto de sus unidades, se obtuvo una prueba topológica de la existencia de una cantidad infinita de ideales maximales. Finalmente, se propone una lista de problemas con el objetivo de establecer una guía para el trabajo futuro sobre este tema.
Motivated by the studies on the topological spaces of Furstenberg and Golomb, the author defined a topology $\tau_M$ on the set of positive integers $\mathbb{N}$, which is generated by the basis $\beta_M := \{\sigma_n : n \in \mathbb{N}\}$ where $\sigma_n := \{m \in \mathbb{N} : \gcd(n,m)=1\}$. The topology $\tau_M$ can be generalized over commutative rings $R$ with $1_R \neq 0_R$ as follows: for each $r \in R \setminus \{0\}$, define $\sigma_r := \{s \in R \setminus \{0\} : \langle r \rangle + \langle s \rangle = R\}$. Hence, the generalization of the topology $\tau_M$ over $R \setminus \{0\}$ is the topology $\tau_{R^0}$ generated by the basis $\beta_R := \{\sigma_r : r \in R \setminus \{0\}\}$. In this work, the topology $\tau_M$ and its generalization over integral domains were studied. Both on $\mathbb{N}$ and over any integral domain $R$, the topological spaces $(\mathbb{N}, \tau_M)$ and $(R \setminus \{0\}, \tau_{R^0})$ are hyperconnected, ultraconnected, and coaser than Golomb spaces. In particular, when $R$ is a field, the topological space $(R \setminus \{0\}, \tau_{R^0})$ is trivial. Over $\mathbb{N}$, it is shown that the set of continuous polynomial functions on $(\mathbb{N}, \tau_M)$ consists of monomials over $\mathbb{N}$. Moreover, it is shown that the set of homeomorphisms on $(\mathbb{N}, \tau_M)$ is the set of bijections $h$, on $\mathbb{N}$, such that for every positive integer $n$, it holds that $h(\sigma_n) = \sigma_{h(n)}$. A topological proof of the infinitude of prime numbers and of the infinitude of primes in the arithmetic progression $4(\mathbb{N} \cup \{0\}) + 3$ is presented. Over infinite integral domains, that are not fields and whose cardinality is greater than that of their unit group, a topological proof of the existence of an infinite number of maximal ideals is obtained. Finally, a list of problems is proposed with the aim of establishing a guide for future work on this topic.
Motivated by the studies on the topological spaces of Furstenberg and Golomb, the author defined a topology $\tau_M$ on the set of positive integers $\mathbb{N}$, which is generated by the basis $\beta_M := \{\sigma_n : n \in \mathbb{N}\}$ where $\sigma_n := \{m \in \mathbb{N} : \gcd(n,m)=1\}$. The topology $\tau_M$ can be generalized over commutative rings $R$ with $1_R \neq 0_R$ as follows: for each $r \in R \setminus \{0\}$, define $\sigma_r := \{s \in R \setminus \{0\} : \langle r \rangle + \langle s \rangle = R\}$. Hence, the generalization of the topology $\tau_M$ over $R \setminus \{0\}$ is the topology $\tau_{R^0}$ generated by the basis $\beta_R := \{\sigma_r : r \in R \setminus \{0\}\}$. In this work, the topology $\tau_M$ and its generalization over integral domains were studied. Both on $\mathbb{N}$ and over any integral domain $R$, the topological spaces $(\mathbb{N}, \tau_M)$ and $(R \setminus \{0\}, \tau_{R^0})$ are hyperconnected, ultraconnected, and coaser than Golomb spaces. In particular, when $R$ is a field, the topological space $(R \setminus \{0\}, \tau_{R^0})$ is trivial. Over $\mathbb{N}$, it is shown that the set of continuous polynomial functions on $(\mathbb{N}, \tau_M)$ consists of monomials over $\mathbb{N}$. Moreover, it is shown that the set of homeomorphisms on $(\mathbb{N}, \tau_M)$ is the set of bijections $h$, on $\mathbb{N}$, such that for every positive integer $n$, it holds that $h(\sigma_n) = \sigma_{h(n)}$. A topological proof of the infinitude of prime numbers and of the infinitude of primes in the arithmetic progression $4(\mathbb{N} \cup \{0\}) + 3$ is presented. Over infinite integral domains, that are not fields and whose cardinality is greater than that of their unit group, a topological proof of the existence of an infinite number of maximal ideals is obtained. Finally, a list of problems is proposed with the aim of establishing a guide for future work on this topic.
Description
Date
2025-06-30
Journal Title
Journal ISSN
Volume Title
Publisher
Collections
Keywords
Integral domains, Golomb topology, Furstenberg topology, Macías topology, Prime numbers